I'm currently reading the book Probability and Random Processes by Grimmett and there is an example that I want to understand, it is the example $3)$ in the section 4.6 Conditional distributions and conditional expectation. To be more precise, it is about the integral below
\begin{aligned} \mathbb{P}\left(X^{2}+Y^{2} \leq 1\right) &=\iint_{A} f_{X, Y}(x, y) d x d y \\ &=\int_{x=0}^{1} f_{X}(x) \int_{y \in A(x)} f_{Y \mid X}(y \mid x) d y d x \\ &=\int_{0}^{1} \min \left\{1, \sqrt{x^{-2}-1}\right\} d x=\log (1+\sqrt{2}) \end{aligned}
where $$ f_{X, Y}(x, y)=\frac{1}{x}, \quad 0 \leq y \leq x \leq 1 . $$
$$ f_{X}(x)=1 \quad \text { if } \quad 0 \leq x \leq 1, \quad f_{Y \mid X}(y \mid x)=\frac{1}{x} \quad \text { if } \quad 0 \leq y \leq x \leq 1, $$
I don't understand how he solved the integral to find $\log(1 + \sqrt{2})$. Has someone an idea? Thanks in advance!